EI laminated iron core transformers...

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Most power tranformer for mains have the EI core, interlaced for better efficiency and solidity.
A non-interlaced EI core also works. Air gap between laminations is kept to minimal in its assembly.

What are the effects of slightly increasing the air gap between core sections ?

What changes are to expect in the voltage, current, heating, power capability of such transformer with an implemented adjustable air gap, in idling and full rated conditions ?
 

Increased magnetizing current, with respective high air gap the magneitzing current would cause additional losses.

Noone would do this intentionally.
 

An air gap doesn't increase the leakage inductance, or if it does only slighty. As a rule of thumb the leakage inductance of a transformer equals the leakage inductance of the air core transformer, thus isn't modified by the air gap.

Welding tranformers and similar designs with artificially increased leakage inductance, e.g. magnetron power supplies, have a magnetical bypass (mechanical variable in case of the welding transformer).
 

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